406 XV NORMED ALGEBRAS AND SPECTRAL THEORY so that Ln (—L) = 0, which is a contradiction. Observe that if AeL, then A2 is an eigenvalue of T* and m(T, A2) = 1.) Show also that, if T is invertible, there exists e > 0 such that every operator Tl e &(E) satisfying || TI - T\\ ^ e is invertible and such that there exists no operator 7" e &(E) for which T2 = TI (use (c).) 10. Let iQ be a bounded open subset of C, and let H be the Hilbert space of analytic functions / on 0 such that if. iy)\*dxdy< -f-oo (cf. Problem in Section 9.13). Let T be the operator which maps each /e H to the function £*-»£/(£)• Show that, for each A e ft and each function g e H such that 0(A) = 0, there exists a unique /eH such that (J-A/) -f=g. Also, if the disk |£ - A| < S is contained in O, then \\g\\* ^ J S2||/||2. Deduce that Sp(T) is the closure H of Q, in C, and that Q is contained in Sp(T) - Spa(r). Deduce from these results that if Q, is taken to be the open annulus rj.<\z\<r2 (where rx > 0), then the operator T is invertible and has no square root in ^f(H) (cf. Problem 9). 11. Let E be a separable Hilbert space, K a compact subset of C, and (x,y)\-+mXty a continuous sesquilinear mapping of E x E into the space MC(K) of complex measures on K. Suppose that and that the measure mXt x is positive for all x e E. Let T be the continuous operator on E such that for all x, ^ in E. For each function /e ^C(K), let/(T) denote the operator defined by for all x, j; in E (Section 15.10, Problem 1). The mapping T-+f(T) of ^C(K) into JSf(E) is linear and such that T* = c(T), where c(£) = f , but this mapping is not in general an algebra homomorphism. Prove that there exists a separable Hilbert space H, the Hilbert sum of E and another Hilbert space F, and a representation /(-»• V(f) of ^c(K) on ^(H) such that, if P is the orthogonal projection of H on E, we have /(T) « P V(f) | E. (Apply Problem 6 of Section 15.9, by taking F to be the set of all finite products of characteristic functions <p*n(n e N) of universally measurable sets in K, chosen in such a way that these functions form a total set in each of the spaces •5fc(K, mXltt *„), where (xn) is a dense sequence in E.) ("Neumark's theorem.") 12, Let E be a separable Hilbert space. Let H be a self-adjoint operator on E such that 0 <£ H^ 1E. Show that there exists a separable Hilbert space G which is the Hilbert sum of E and a Hilbert space F, and an orthogonal projector g on G such that H— PQ \ E, where P is the orthogonal projection of G on E. (For x9 y in E, define mXt y to be the measure carried by the set of two points {0, 1} such that mx, y(W) = (On - ff) ' x \ y) and mx, y({l}) = (H • x \ y\ and apply Problem 11.)r e>0 such that m(T, A) = m(TQ, A0)